Abstract
Decision-theoretic representation theorems have been developed and appealed to in the service of two important philosophical projects: (a) in attempts to characterise credences in terms of preferences, and (b) in arguments for probabilism. Theorems developed within the formal framework that Savage developed have played an especially prominent role here. I argue that the use of these ‘Savagean’ theorems create significant difficulties for both projects, but particularly the latter. The origin of the problem directly relates to the question of whether we can have credences regarding acts currently under consideration and the consequences which depend on those acts; I argue that such credences are possible. Furthermore, I argue that attempts to use Jeffrey’s non-Savagean theorem (and similar theorems) in the service of these two projects may not fare much better.
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Notes
More specifically, the question here concerns the conditions under which an agent counts as being in such-and-such a credence state. The orthodoxy is that belief is a binary relation between a subject at a time and a proposition, and most I imagine would be happy to extend this approach to credences: a credence of n towards P is a ternary relation between a subject at a time, a degree of confidence (represented by n), and a proposition P. The real philosophical meat lies in specifying the conditions under which an agent stands in such a relationship.
What is it for \( {\mathcal{C}}r \) to be an accurate model of a subject's credences? I have intentionally left this matter somewhat vague to allow for variation amongst individual theorists. (Some of this variation will be discussed further in Sect. 3). The question hinges in part upon one’s metaphysics of the graded attitudes, and on one’s account of just how—and to what extent—credence and utility functions are supposed to faithfully represent those attitudes.
That the probability function derived using Savage’s theorem in particular does not directly supply credence values for acts has been noted before (e.g., Spohn 1977, pp. 117–118, Joyce 1999, p. 117), though the relevance of the point for the theorem’s use in arguments for probabilism and as a basis for preference functionalism has not been discussed.
A referee points out that whenever a theorem T establishes the existence of a probabilistic \( {\mathcal{C}}r \) and a \( {\mathcal{U}} \) relative to which ≽ maximises EU, where \( {\mathcal{C}}r \) is defined on \( \varvec{\mathcal{E}} \) and \( \varvec{\mathcal{E}} \) does not contain all objects of credence, then there will be a simple ‘extension’ of the theorem, T*, according to which a probabilistic \( {\mathcal{C}}r \)* exists relative to an appropriately expanded set of propositions (call it \( \varvec{\mathcal{E}} \)*, which we’ll assume has an algebraic structure) which (a) agrees with \( {\mathcal{C}}r \) on \( \varvec{\mathcal{E}} \) and (b) combines with \( {\mathcal{U}} \) to represent ≽ in just the same way as \( {\mathcal{C}}r \) did. We might then try to re-run the argument using T*, and avoid the problems that I will raise later with P4.
There are two points to note about this. First, for the reasons to be discussed, ≽ on \( \varvec{\mathcal{A}} \) will palce no interesting constraints on \(\mathcal{C}r^{*}\) over \( \varvec{\mathcal{E}}^{*} \text{-}\varvec{\mathcal{E}} \), so there will be many probabilistic \( {\mathcal{C}}r^{*} \) satisfying (a) and (b), including ones which disagree with respect to the confidence ranking. Something would need to be said about the very significantly weakened uniqueness conditions here. Second, and more importantly, there will also be many non-probabilistic \( {\mathcal{C}}r \)* which also satisfy (a) and (b). In the context of an argument for probabilism, we cannot assume that only the probabilistic \( {\mathcal{C}}r \)* can form legitimate representations of ideally rational agents’ credences. More generally: if we re-state the argument using T*, then while it may be plausible that \( {\mathcal{C}}r \)*’s domain covers all the required objects of credences, we have at best only shifted the bump under the rug.
It’s worth pointing out that there are other representation theorems with stronger uniqueness results than Savage’s. For instance, Ramsey’s (1931) theorem establishes not only that there’s a unique probability function \( {\mathcal{C}}r \) which figures in an expected utility representation of ≽, but also that \( {\mathcal{C}}r \) is the only function into ℝ with this property. (The form of the representation is not identical to the EU function defined in Sect. 2, but is recognisably an expected utility formula.) See Elliott (forthcoming) for discussion and relevant proofs. While Ramsey’s theorem is not ordinarily cast within the Savagean framework, it is straightforward to do so without any changes to the proof.
These are hardly the only problems that Meacham and Weisberg point to, and there are more problems still in the wider literature. Again: a thorough defence of preference functionalism is not the point here, nor would such be possible in the available space.
Thanks to Alan Hájek for this way of putting the point.
Jeffrey does require that \( \varvec{\mathcal{P}} \), minus a special set of \( \varvec{\mathcal{N}} \) of ‘null’ propositions, is atomless. This implies that \( \varvec{\mathcal{P}} \) − \( \varvec{\mathcal{N}} \) cannot contain all subsets of \( \varvec{\mathcal{W}} \) (e.g., it can’t contain singleton sets), but any other proposition can be placed in \( \varvec{\mathcal{N}} \) and assigned a credence of 0 in the final representation.
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Acknowledgments
Thanks to Rachael Briggs, David Chalmers, Alan Hájek, Jessica Isserow, Leon Leontyev, Hanti Lin, J. Robert G. Williams, and several anonymous referees for helpful discussion and feedback. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007–2013)/ERC Grant Agreement No. 312938.
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Elliott, E. Probabilism, Representation Theorems, and Whether Deliberation Crowds Out Prediction. Erkenn 82, 379–399 (2017). https://doi.org/10.1007/s10670-016-9824-8
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DOI: https://doi.org/10.1007/s10670-016-9824-8